Quadratic Relations

The Basics of an Advanced Math

History, Information and Use of Quadratics

Quadratics were discovered by the Babylonians around 2000 BC. Quadratics can be used to figure out the flight path of an object. For example, you can use it to figure out the highest point and the exact spot an object can land at if it were launched from a catapult. Quadratics are also used in the business field to figure out maximum profit. There are many more uses of quadratics but these are the most common ones.

What You Will be Discovering!

Introduction:
  • Properties of a Parabola
  • Second Differences
  • Transformations


Vertex Form: a(x-h)^2+k

  • Finding an Equation with the Vertex and Point Only
  • Isolating for "x"
  • Graphing Vertex Form


Factored Form: a(x-r)(x-s)

  • Expanding Factored to Standard
  • Common Factoring
  • Factoring Simple Trinomials
  • Factoring Complex Trinomials
  • Perfect Square Trinomials
  • Difference of Squares


Standard Form: ax^2+bx+c

  • Standard Form to Vertex Form (Completing the Square)
  • Discriminants
  • Solving using Quadratic Equation
  • Solving using Quadratic Formula


Word Problems:

  • Geometry Problem
  • Revenue Problem


Reflection

  • Test
  • Unit Reflection

INTRODUCTION

Properties of a Parabola

The graphed equations are called a parabola. The listed are important parts of a parabola:

  • Vertex: Where the optimal value and the axis of symmetry meet
  • Optimal Value: The highest or the lowest point of the parabola
  • Axis of Symmetry: A vertical line that goes through the middle of the curve splitting it into two equal halves
  • X- Intercepts/Roots/Zeroes: The points where the curve goes through the x-axis. (All real numbers)
  • Y- Intercept: The point where the curve goes through the y-axis. (Y below optimal value or above optimal value)
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Second Differences

Second differences are used to check if a relation is linear or quadratic. Making a table of second differences makes it easy to figure out the relation by just looking at the numbers on the table. Quadratic relations always have an equal second difference, however an unequal first difference. You can tell the difference between a quadratic and linear relation easily. You should always remember that a linear relation always has an equal first difference and and a quadratic relation does not. In the image below, you can tell it is a quadratic relation because the first difference (which is in green) is unequal, while the second difference (which is in purple) is.
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Transformations

When an equation for a parabola is written in vertex form, each letter (a,h,k) is responsible for changing the form of the parabola:



  • A: Stretches or compresses the parabola. If A is less than one, it stretches. If it's more than one, it compresses.
  • H: Moves the parabola left or right. If H is negative, it moves to the right. If it's positive, it moves to the left.
  • K: Moves the vertex of the parabola up or down. if K is negative, it moves down. If it's positive, it moves up.



The vertex of a transformed parabola is written as (h,k). Watch the video below by Khan Academy for a better understanding.

Shifting and scaling parabolas

Vertex Form

As explained in the video above, the equation for vertex form is written as a(x-h)^2+k. The letters a,h,k are responsible for the transformation of the parabola while the x identifies as a variable. Let's review what each letter (a,h,k) is responsible for:



  • A: Stretches or compresses the parabola. If A is less than one, it compresses. If it's more than one, it compresses.
  • H: Moves the parabola left or right. If H is negative, it moves to the right. If it's positive, it moves to the left.
  • K: Moves the vertex of the parabola up or down. if K is negative, it moves down. If it's positive, it moves up.

Isolating for x

When isolating for x, you will be given an equation in a(x-h)^2+k form. The letters a,h,k will be replaced by a number, leaving x. You will need to isolate x to solve the equation. Here are the steps:


  1. Find y-intercept by setting x=0 and then solve for y. Whatever your y will be the y value for your vertex written as (0,y)
  2. Find your zeroes by setting y=0. Move the k to the other side and change the sign. Divide both sides with the a value. This will allow you to cancel out the a value on both sides. In order to cancel out the exponent, you will have to square root both sides. Once it is square rooted, you will have to bring h to the other side, isolating the x and you have found your zeroes. Your zeroes will be written as (x,0)

Graphing a Parabola in Vertex Form

When graphing in vertex form, always remember that the original equation is a(x-h)^2+k.

View the video below from Khan Academy for the steps to graphing a parabola in vertex form.

Graphing a parabola in vertex form

Writing an Equation Given a Point and the Vertex

As long as a point (x,y) and the vertex (h,k) of a parabola is given, it is fairly easy to find the equation of the parabola. Here are the steps:



  1. Plug in the (x,y) and (h,k) values in their rightful places in the formula
  2. Simplify the equation using BEDMAS
  3. Isolate a one by one by moving the x,y,h,k values to the other side
  4. To isolate a, divide each side by the number without the variable to isolate a and you have found the value of a which will allow you to write the complete equation
Finding the Equation of a Parabola Given a Vertex and a Point

Factored Form

Factors and Zeroes

A zero/root of a parabola is another name for the x- intercepts. To find the x- intercepts, you must set y= 0.


Key points:

  • When the (a) value changes, the zeroes do not change.
  • When the (a) value changes, the axis of symmetry do not change.
  • When the (a) value changes, the optimal value does change.

Expanding (Factored to Standard)

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Common Factoring

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Factoring Simple Trinomials

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Factoring Complex Trinomials

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Difference of Squares

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Perfect Square

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Standard Form

The equation for standard form is written as ax^2+bx+c=0. The a,b and c values are known values that are plugged into this equation. Just remember that a can never be zero or without a value. The value for a must always be 1 or more. The x is the variable which is unknown so we don't know the value until we completely solve the equation. We can use an equation or a formula to completely solve this expression

Quadratic Equations

A quadratic equation is any equation in the form ax^2+bx+c=0. An equation is any mathematical, or in this case, quadratic statement in which two expressions are equal to each other. Quadratic equations can be solved by factoring, which you have already learned. When a quadratic equation is solved, you find the zeroes of the equation which will help you graph the equation. There are new ways that are going to be introduced in which a quadratic equation can be solved. The two ways that you are going to learn are completing the square and the quadratic formula.

Completing the Square (Standard to Vertex)

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The Quadratic Formula

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Discriminants

You can find the discriminant inside of the quadratic formula. The discriminant is found in the form of +b^2-4ac inside of the square root. Basically, the discriminant is the number located inside of the square root in the formula. From the discriminant, you can determine how many solutions (roots/zeroes) a quadratic equation has. Here is how you can tell how many solutions an equation could have:



  • If +b^2-4ac = more than zero, there are 2 solutions.
  • If +b^2-4ac = zero, there is only one solution.
  • If +b^2-4ac = less than zero, there is no solution.

Optimization Problem

You have 250 meters of fence. You need to fence of three sides of a garden, with the fourth side being against a building. What dimensions would provide the maximum area and what is the maximum area?
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Motion Problem

A ball is thrown upward at an initial velocity of 8.4 m/s, form a height of 12 meters above ground. The height of the ball (m), above the ground after t seconds is modelled by the equation h=-4.9t^2+8.4t+1.2.


a) How long will it take the ball to fall to the ground?

b) What is the maximum height of the ball and at what time will it reach this height?


*Round answers to the nearest tenth when working with decimals*

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Revenue Problem

Calculators are sold to students for 20 dollars each. 300 students are willing to buy them at that price. For every 5 dollar increase in price, there are 30 fewer students willing to buy calculators. What selling price will produce the maximum revenue and what will the maximum revenue be?
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Test: My Lowest Mark

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Unit Reflection

The unit of quadratics was the largest unit in math this semester. I believe that this was definitely one of the toughest units and was very hard to keep up with since there was so much information to take in. As you can see the test above, it was my worst performance on a test. I felt like I understood the concept and methods of the lessons but just had a hard time executing everything. During lessons, I felt like I understood everything, but when it came to tests, I felt like I forgot everything I learned in class. In this unit, I struggled the most in application such as graphing parabolas from a specific form of an expression. There were only 3 types of forms which were standard form, vertex form and factored form, but there was a lot of information to learn about each form. Whenever we switched forms, it would confuse me. I am going to admit that creating this website has helped me comprehend the unit a lot more. Since I am now applying my knowledge to this website, it has given me practice. This is why I found the Quadratics Part 3 Test quite easy. I knew what methods to use, how to use them and what parts of the method are critical to solve the problems given. After studying for the last test in the unit, I found switching from form to form much easier. The quadratic formula was a life saver in this unit and was very simple to solve a quadratic equation with it. I believe that it is easier than factoring. I also find using discriminants very helpful since it informs you how many solutions there are in an equation. Overall, I found this unit very difficult but with practice, I have learned methods and ways to successfully overcome this unit!

Amazing Resources!

Khan Academy

Math Lessons Made Easy

Desmos Graphing Calculator

Beautiful, Free Math